where M(2, C) denotes the set of 2 by 2 complex matrices. By considering C2diffeomorphic to R4 and M(2, C) diffeomorphic to R8 we can see that φ is an injective real linear map and hence an embedding. Now, considering the restriction of φ to the 3-sphere (since modulus is 1), denoted S3, we can see that this is an embedding of the 3-sphere onto a compact submanifold of M(2, C). However it is also clear that φ(S3) = SU(2). Therefore as a manifold S3 is diffeomorphic to SU(2) and so SU(2) is a compact, connected Lie group.

The above representation bases generalize to n > 3. The Lie algebra corresponding to SU(n) is denoted by su(n). Its standard mathematical representation consists of the tracelessantihermitiann×n complex matrices, with the regular commutator as Lie bracket. A factor i is often inserted by particlephysicists, so that all matrices become Hermitian. This is simply a different, more convenient, representation of the same real Lie algebra. Note that su(n) is a Lie algebra over R.

If we choose an (arbitrary) particular basis, then the subspace of traceless diagonaln×n matrices with imaginary entries forms an (n − 1)-dimensional Cartan subalgebra.

Complexify the Lie algebra, so that any traceless n×n matrix is now allowed. The weighteigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal. Even though the Cartan subalgebra h is only (n − 1)-dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only. So, we have a basis where the i-th basis vector is the matrix with 1 on the i-th diagonal entry and zero elsewhere. Weights would then be given by n coordinates and the sum over all n coordinates has to be zero (because the unit matrix is only auxiliary).

So, SU(n) is of rankn − 1 and its Dynkin diagram is given by An−1, a chain of n − 1 vertices. Its root system consists of n(n − 1) roots spanning a n − 1Euclidean space. Here, we use n redundant coordinates instead of n − 1 to emphasize the symmetries of the root system (the n coordinates have to add up to zero). In other words, we are embedding this n − 1 dimensional vector space in an n-dimensional one. Then, the roots consists of all the n(n − 1) permutations of (1, −1, 0, ..., 0). The construction given two paragraphs ago explains why. A choice of simple roots is

Often one will see the notation SU(p, q) without reference to a ring or field; in this case, the ring or field being referred to is C and this gives one of the classical Lie groups. The standard choice for A when F = C is

However there may be better choices for A for certain dimensions which exhibit more behaviour under restriction to subrings of C.

In physics the special unitary group is used to represent bosonic symmetries. In theories of symmetry breaking it is important to be able to find the subgroups of the special unitary group. Subgroups of SU(n) that are important in GUT physics are, for p > 1, n − p > 1 :

Since the rank of SU(n) is n − 1 and of U(1) is 1, a useful check is that the sum of the ranks of the subgroups is less than or equal to the rank of the original group. SU(n) is a subgroup of various other Lie groups: